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Glogic Irfan - PhD thesis.pdf (624.18 KB)

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On the existence and stability of self-similar blowup in nonlinear wave equations

Glogic, Irfan
http://orcid.org/0000-0002-9382-9293
http://rave.ohiolink.edu/etdc/view?acc_num=osu1523692938316794

Abstract Details

2018, Doctor of Philosophy, Ohio State University, Mathematics.
We study the existence and stability of singularity formation in nonlinear wave equations. First, we formulate the problem of mode stability of self-similar blowup solutions to a class of nonlinear wave equations. We then develop a method of proving mode stability in this context and illustrate it on two important examples, the spherically symmetric Yang-Mills equation and equivariant wave maps into a sphere, for which the problem was open for almost a decade. Our method is broadly applicable and provides a general approach to stability problems related to self-similar solutions of nonlinear wave equations. This part of the thesis is based on two research papers that already appeared in print, see [13,16]. Next, we study the singularity formation in wave maps into a negatively curved target manifold. More precisely, we consider wave maps on (1+d)-dimensional Minkowski space. For each dimension d≤8 we construct a negatively curved, d-dimensional target manifold that allows for the existence of a self-similar wave map which provides a stable blowup mechanism for the corresponding Cauchy problem. In particular, we illustrate how the problem of mode stability plays a decisive role in the proof of the (nonlinear) stability of self-similar blowup. This part is based on a joint work with Roland Donninger [19].
Ovidiu Costin (Advisor)
Saleh Tanveer (Committee Member)
Barbara Keyfitz (Committee Member)
107 p.
Mathematics

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Citations

  • Glogic, I. (2018). On the existence and stability of self-similar blowup in nonlinear wave equations [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1523692938316794

    APA Style (7th edition)

  • Glogic, Irfan. On the existence and stability of self-similar blowup in nonlinear wave equations. 2018. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1523692938316794.

    MLA Style (8th edition)

  • Glogic, Irfan. "On the existence and stability of self-similar blowup in nonlinear wave equations." Doctoral dissertation, Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu1523692938316794

    Chicago Manual of Style (17th edition)

osu1523692938316794
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This open access ETD is published by The Ohio State University and OhioLINK.